An extension of the Beurling-Chen-Hadwin-Shen theorem for noncommutative Hardyspaces associated with finite von Neumann algebras

Cupples I Room 207

In 2015, Yanni Chen, Don Hadwin and Junhao Shen proved a noncommutative version of Beurling's theorem for a continuous unitarily invariant norm α on a tracial von Neumann algebra (M,τ)such that α is one dominating with respect to τ. The role of H^∞ is played by a maximal subdiagonal algebra A . In the talk, we first will show that if α is a continuous normalized unitarily invariant norm on (M,τ), then there exists a faithful normal tracial state ρ on M and a constant c >0 such that α is a c times one norm-dominating norm on (M,ρ). Moreover, ρ (x)= τ (xg), where x in M, g is positive in L^1 (Z,τ), where Z is the center of M . Here c and ρ are not unique. However, if there is a c and ρ so that the Fuglede-Kadison determinant of g is positive, then Beurling-Chen-Hadwin-Shen theorem holds for L^(α ) (M,τ). The key ingredients in the proof of our result include a factorization theorem and a density theorem for for L^(α ) (M,ρ).